Vitali's coverage theorem

The coverage rate of Vitali is a set of measure theory , a subdivision of mathematics that deals with the generalization of length, area and volume terms. The theorem is an aid for proving that for the Lebesgue-Stieltjes measure the Radon-Nikodým derivative (with respect to the Borel measure ) and the ordinary derivative agree. The sentence is named after Giuseppe Vitali , who proved it in 1908.

Framework

It denotes the Lebesgue measure and the outer Lebesgue measure, i.e. the outer measure that is generated by the Lebesgue pre-measure . An amount family of open, closed or half-open intervals with is a Vitali-coverage of a (not necessarily measurable ) amount if, for all all and a exists, so that and . ${\displaystyle \lambda }$ ${\displaystyle \eta }$ ${\displaystyle {\mathcal {V}}}$ ${\displaystyle I}$ ${\displaystyle \lambda (I)>0}$ ${\displaystyle A\subset \mathbb {R} }$ ${\displaystyle x\in A}$ ${\displaystyle \varepsilon >0}$ ${\displaystyle I\in {\mathcal {V}}}$ ${\displaystyle x\in I}$ ${\displaystyle \lambda (I)<\varepsilon }$

statement

If for an arbitrary set with a Vitali cover is given, then for each there is a finite number of disjoint intervals in such that ${\displaystyle A\subset \mathbb {R} }$ ${\displaystyle \eta (A)<\infty }$ ${\displaystyle {\mathcal {V}}}$ ${\displaystyle \varepsilon >0}$ ${\displaystyle I_{1},I_{2},\dots ,I_{n}}$ ${\displaystyle {\mathcal {V}}}$

${\displaystyle \eta \left(A\setminus \bigcup _{i=1}^{n}I_{n}\right)<\varepsilon }$

applies.

Web links

• IA Vinogradova: Vitali theorem . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).

literature

• Jürgen Elstrodt: Measure and integration theory . 6th, corrected edition. Springer-Verlag, Berlin Heidelberg 2009, ISBN 978-3-540-89727-9 , doi : 10.1007 / 978-3-540-89728-6 .